Equal BER power control for uplink MC-CDMA with MMSE successive interference cancellation

ABSTRACT

For a given decision order, MMSE successive interference cancellation (MMSE-SIC) can simultaneously maximize SIRs of all users. To further increase its efficiency, a power control (PC) algorithm, under equal BER criterion, is disclosed for uplink MC-CDMA. In frequency-selective Rayleigh fading channels, the MMSE-SIC integrated with the equal BER PC suppresses multiple access interference (MAI) effectively, resulting in a performance very close to the single user bound (SUB).

RELATED APPLICATIONS

This application claims priority from U.S. Provisional Patent application Ser. No. 60/574,191, filed May 25, 2004.

GOVERNMENT LICENSE RIGHTS

The United States government may hold license and/or other rights in this invention as a result of financial support provided by governmental agencies in the development of aspects of the invention.

FIELD OF INVENTION

This invention relates generally to CDMA systems, and more specifically relates to a method for efficiently suppressing multiple access interferences (MAI), a major factor limiting the capacity of uplink MC-CDMA systems.

BACKGROUND OF INVENTION

The performance of CDMA systems is limited by multiple access interference (MAI). Among many interference cancellation schemes, successive interference cancellation (SIC) is highly desirable, due to its low complexity, high compatibility with existing systems and easy accommodation to strong error-correcting codes [A. J. Viterbi, “Very low rate convolutional codes for maximum theoretical performance of spread-spectrum multi-access channels,” IEEE J. Select Areas Commun. Vol. 8, pp. 641-649, May 1990]. However, unlike other detection techniques, SIC is sensitive to received power allocation. By providing channel state information (CSI) at the receiver and reliable feedback of power allocation from the receiver to the transmitter, we are able to integrate SIC with power control (PC), which can improve system capacity significantly.

For a system which aims to achieve comparable performance for all users, equal BER criterion is suitable for deriving the power allocation. As has been concluded in the literature, equal BER PC benefits SIC significantly by increasing the reliability of earlier detected users. Nevertheless, most of the work focused on (match filter) SIC (MF-SIC) [Viterbi op. cit.; G. Mazzini “Equal BER with successive interference cancellation DS-CDMA systems on AWGN and Ricean channels,” in Proc. ICCC PIMRC, July 1995, pp. 727-731; R. M. Buehrer, “Equal BER performance in linear successive interference cancellation for CDMA systems,” IEEE Trans. Commun., vol. 49, no. 7, pp 1250-1258, July 2001]. With the increase of system load, in CDMA systems, the performance of MF degrades quickly, limiting the effectiveness of SIC. Therefore, it is meaningful to integrate PC with SIC for more powerful detection techniques, such as decorrelating and MMSE. For a given decision order, MMSE SIC (MMSE-SIC) maximizes all users' SIRs simultaneously [T. Guess, “Optimal sequences for CDMA with decision-feedback receivers,” IEEE Trans. Commun., vol., 49, pp. 886-900, April 2003]. Therefore, in this invention we consider the equal BER PC algorithm for this optimal SIC receiver in quasi-synchronous uplink MC-CDMA.

SUMMARY OF INVENTION

For a given decision order, MMSE successive interference cancellation (MMSE-SIC) can simultaneously maximize SIRs of all users [T. Guess Op. cit.]. To further increase its efficiency, a power control (PC) algorithm, under equal BER criterion, is used in this invention for uplink MC-CDMA. In frequency-selective Rayleigh fading channels, the MMSE-SIC integrated with the equal BER PC suppresses multiple access interference (MAI) effectively, resulting in a performance very close to the single user bound (SUB). In the present invention a method is thus disclosed for efficiently suppressing multiple access interference (MAI), a major factor limiting the capacity of uplink MC-CDMA systems. A novel power control algorithm is used under equal BER criterion for a nonlinear MMSE-SIC receiver.

BRIEF DESCRIPTION OF DRAWINGS

The invention is illustrated by way of example in the drawings appended hereto in which:

FIG. 1 is a schematic block diagram of an MC-CDMA system in accordance with the invention, with MMSE receiver integrated with the equal BER PC;

FIG. 2 is a graph of BER performance, with different receiver structures, with and without PC, over 16 users versus the average E_(b)/N₀ per user; and

FIG. 3 is a graph of BER performance, with different receiver structures, with and without PC, over 16 users versus the average E_(b)/N₀ per user.

DESCRIPTION OF PREFERRED EMBODIMENT

In FIG. 1 a block diagram of the MC-CDMA system with MMSE-SIC receiver integrated with the equal BER PC is schematically depicted. Referring to the Figure, the method comprises the following steps:

(a) Based on the channel state information (CSI) obtained at the receiver, the “Equal BER Power Control” block is employed to calculate the transmit power allocation of different users. A successive algorithm is used, which searches the transmit power of different users under Equal BER criterion with a total transmit power constraint (the “Multi-carrier Channel Model” represents a concatenation of IDFT, wireless fading channel and DFT).

(b) With the assumption of slow fading channel, the calculated power allocation is fed back to the transmitter so that each user will transmit with the assigned power. (b denotes a vector including transmit symbols of all users.)

(c) At the receiver, the non-linear MMSE-SIC receiver is employed. (The block diagram of FIG. 1 shows a standard operation of the non-linear MMSE-SIC receiver. The output of DFT x is first processed with a Matched-Filter Bank. Then, the output y is processed by a feedforward matrix F. After that, based on a certain decision order and output z, a “Hard Decision Device” is used to make decisions on certain transmit symbols with MMSE criterion and the earlier detected symbols {circumflex over (b)} are fed back through a feedback matrix B to assist in detecting other symbols.) The equal BER power control ensures that different users achieve the same signal-to-interference (SIR) ratio after SIC, hence, significantly improving the performance of SIC and effectively suppressing MAI.

By properly defining the search district and with some well-known search algorithms, only a small number of searches are required for each channel realization. Therefore, this power control algorithm has a low complexity, particularly under a slow fading channel.

Simulation results show that the MAI can be suppressed effectively, resulting in a performance very close to the theoretical limit MMSE-SIC Receiver for MC-CDMA.

In quasi-synchronous uplink MC-CDMA, with total N sub-carriers and K active users, for the k^(th) user, each transmit symbol is replicated into copies and each copy is multiplied by a chip of a preassigned spreading code c_(k) of length N (frequency domain spreading). After transforming by an N-point IDFT and parallel-to-serial (P/S) conversion, a cyclic prefix (CP) is inserted between successive OFDM symbols to avoid inter-symbol interference (ISI). Finally, after RF upconversion, the signal is transmitted through the channel [S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., Vol. 35, no. 12, pp. 126-133, December 1997].

A frequency-selective Rayleigh fading channel is considered. However, with the use of CP, the channel can be considered frequency-nonselective over each sub-carrier [Z. Wang and G. B. Giannakis, “Wireless multicarrier communications where Fourier meets Shannon,” IEEE Signal Processing Mag., pp. 29-48, May 2000]. We assume time-invariant during each OFDM symbol, hence, the channel for the k^(th) user can be represented by an (N×1) vector, h_(k)=[h_(k,1), h_(k,2), . . . , h_(k,N)]^(T), where each element is a complex Gaussian random variable with unit variance. Furthermore, due to the proximity and partial overlap of signal spectra, correlated fading on different sub-carriers is considered. The correlation between two sub-carriers depends on their frequency spacing and the RMS channel delay spread τ_(d) [W. C. Jackes, Microwave Mobile Communications. New York: Wiley, 1974].

After discarding the CP, the received signal is demodulated by an N-point DFT, and the output during the i^(th) OFDM symbol interval can be expressed in a compact matrix form as x(i)={tilde over (C)}Ab(i)+η(i), where {tilde over (C)}=[h₁·c₁, h₂·c₂, . . . , h_(K)·c_(K)] denotes the channel-modified spreading code matrix, with · representing element-wise multiplication; A=diag(a₁, a₂, . . . , a_(K)) is a diagonal matrix containing the received amplitudes of all users and b(i)=[b₁(i), b₂(i), . . . , b_(K)(i)]^(T) containing all parallel transmitted symbols, which are assumed BPSK modulated with normalized power; The (N×1) white Gaussian noise vector η(i) has zero mean and covariance matrix σ²I, where I is an (N×N) identity matrix. After match filtering, we have y(i)={tilde over (C)} ^(H) ·x(i)=RAb(i)+{tilde over (η)}(i), where R={tilde over (C)}^(H){tilde over (C)} is the channel-modified cross correlation matrix. The MMSE-SIC receiver is implemented using the Cholesky factorization (CF) of the positive definite matrix R_(m)=R+σ²A⁻², which can be uniquely decomposed as R_(m)=Γ^(H)D²Γ, with Γ upper triangular and monic (having all ones along the diagonal) and D²=diag([d₁ ², d₂ ², . . . , d_(K) ²]^(T)) having positive elements on its diagonal. Multiplying on both sides of equation (yy) by D⁻²Γ^(−H), we obtain z(i)=D ⁻²Γ^(−H) ·y(i)=ΓAb(i)+{circumflex over (η)}(i), where {circumflex over (η)}(i) is a (K×1) vector with uncorrelated components, (Note that the extra term −D⁻²Γ^(−H)σ²A⁻¹b(i) was included into {circumflex over (η)}(i).) whose covariance matrix R=_({circumflex over (η)}(i))=σ²D⁻² [G. Ginis and J. Cioffi, “On the relationship between V-BLAST and the GDFE,” IEEE Commun. Lett., vol. 5, pp. 364-366, September 2001]. Since Γ is upper triangular and {circumflex over (η)}(i) has uncorrelated components, b(i) can be recovered by back-substitution combined with symbol-by-symbol detection. The detection algorithm is as follows, for  k = 0  to  K − 1 ${{\hat{b}}_{K - k}(i)} = {{hard}\quad{{decision}\left( {\left( {z_{K - k}(i)} \right) - {\sum\limits_{m = 1}^{k}{{a_{K - {k_{+}m}} \cdot \Gamma_{{K - k},{K - {k_{+}m}}}}{{\hat{b}}_{K - {k_{+}m}}(i)}}}} \right)}}$ By ignoring decision errors (It is pointed out in [Guess op. cit.] that in uncoded systems, the effects of error propagation can for the most part be mitigated, when the users are detected in decreasing order of SIR) the SIR of the (k+1)^(th) detected symbol {circumflex over (b)}_(K-k)(i) can be expressed as [G. K. Kaleh, “Channel equalization for block transmission systems,” IEEE J. Select Areas Commun., vol. 13, pp. 110-121, January 1995]: ${SIR}_{K - k} = {{\frac{E\left\lbrack {{a_{K - k}{b_{K - k}(i)}}}^{2} \right\rbrack}{mmse} - 1} = {\frac{a_{K - k}^{2}}{\sigma^{2}d_{K - k}^{- 2}} - 1.}}$ Moreover, when all interferences are cancelled, the last detected symbol {circumflex over (b)}₁(i) achieves the single user bound (SUB), given by ${{BER}_{SUB} = {E_{H}\left\lbrack {Q\left( \sqrt{\frac{a_{1}^{2} \cdot \left( {\frac{1}{N}{\sum\limits_{n = 1}^{N}{h}_{1,n}^{2}}} \right)}{\sigma^{2}}} \right)} \right\rbrack}},$ where E_(H)[·] denotes the expectation over all channel realizations and Q(·) represents the tail of the error function.

Equal BER PC Algorithm

From (snr), to achieve the same BER, for all users, we need a _(K) ² d _(K) ² =a _(K-1) ² d _(K-1) ² = . . . =a ₁ ² d ₁ ². Expressing R_(m)=R+σ²A⁻² and its CF, R_(m)=Γ^(H)D²Γ in details, we get the following two equal matrices, ${\begin{bmatrix} {r_{1,1} + {\sigma^{2}a_{1}^{- 2}}} & r_{1,2} & \ldots & r_{1,K} \\ r_{2,1} & {r_{2,2} + {\sigma^{2}a_{2}^{- 2}}} & \ldots & r_{2,k} \\ \vdots & \vdots & ⋰ & \vdots \\ r_{K,1} & r_{K,2} & \ldots & {r_{K,K} + {\sigma^{2}a_{K}^{- 2}}} \end{bmatrix}\quad{{and}\begin{bmatrix} d_{1}^{2} & {d_{1}^{2}\gamma_{1,2}} & \ldots & {d_{1}^{2}\gamma_{1,K}} \\ {d_{1}^{2}\gamma_{1,2}^{*}} & {\sum\limits_{k = 1}^{2}{d_{k}^{2}{\gamma_{k,2}}^{2}}} & \ldots & {\sum\limits_{k = 1}^{2}{d_{k}^{2}\gamma_{k,K}\gamma_{k,2}}} \\ \vdots & \vdots & ⋰ & \vdots \\ {d_{1}^{2}\gamma_{1,K}^{*}} & {\sum\limits_{k = 1}^{2}{d_{k}^{2}\gamma_{k,K}^{*}\gamma_{k,2}}} & \ldots & {\sum\limits_{k = 1}^{K}{d_{k}^{2}{\gamma_{k,K}}^{2}}} \end{bmatrix}}},$ where * denotes complex conjugate, r_(i,j) and γ_(i,j) denote the (i,j)^(th) element of R and Γ, respectively. Notice γ_(i,j)=1 when i=j. Since R_(m) is Hermitian symmetric, we only consider the lower triangle. Defining a_(k) ²d_(k) ²

λ, then (snr) becomes ${SIR}_{k} = {\frac{\lambda}{\sigma^{2}} - 1}$ (k=1, 2, . . . , K), which greater than zero for λ>σ². By equating the first column of (m1) and (m2), we obtain the following K equations $\left\{ {\begin{matrix} {{r_{1,1} + {\sigma^{2}a_{1}^{- 2}}} = d_{1}^{2}} \\ {r_{2,1} = {d_{1}^{2}\gamma_{1,2}^{*}}} \\ \vdots \\ {r_{K,1} = {d_{1}^{2}\gamma_{1,K}^{*}}} \end{matrix}.} \right.$ Substituting $d_{1}^{2} = \frac{\lambda}{a_{1}^{2}}$ into the first equation of (e1), we get $a_{1}^{2} = {{\frac{\lambda - \sigma^{2}}{r_{1,1}}\quad{and}\quad d_{1}^{2}} = {\frac{\lambda\quad r_{1,1}}{\lambda - \sigma^{2}}.}}$ Applying d₁ ² in the rest equations, we obtain $\gamma_{1,k} = \frac{r_{k,1}^{*}}{d_{1}^{2}}$ (k=2, 3, . . . , K). Similarly, from the K−1 equations of the second column, we get $a_{2}^{2} = {\frac{\lambda - \sigma^{2}}{r_{2,2} - {{\gamma_{1,2}}^{2}d_{1}^{2}}}.}$ With $d_{2}^{2} = \frac{\lambda}{a_{2}^{2}}$ and the results obtained from the first column, γ_(2,k) (k=3, 4, . . . , K) can be solved. Applying the same method successively for the rest columns, finally, we obtain the power allocation a_(k) ², which can be expressed in the general successive form as $\left\{ \begin{matrix} {a_{1}^{2} = \frac{\lambda - \sigma^{2}}{r_{1,1}}} \\ {a_{k}^{2} = {\frac{\lambda - \sigma^{2}}{r_{k,k} - {\sum\limits_{j = 1}^{k - 1}{{\gamma_{j,k}}^{2}a_{j}^{- 2}\lambda}}}{\left( {{k = 2},\ldots\quad,K} \right).}}} \end{matrix}\quad \right.$ From (result), a_(k) ² (k=1, 2, . . . , K) is a function of λ, is which was proven in Appendix A to satisfy the following property: a_(k) ²ε[0,+∞) (k=1, 2, . . . , K) are monotonically increasing with λε[σ²,+∞). With the above conclusion, under a power constraint Pε[0, +∞), there uniquely exists a (λ)^(†), and with (result), a unique power distribution (a_(k) ²)^(†), which satisfies $\mathcal{P} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{\left( a_{k}^{2} \right)^{\dagger}.}}}$ In conclusion, the algorithm can be described as follows: 1) let λ=σ² 2) applying (result), calculate $\frac{1}{K}{\sum\limits_{k = 1}^{K}{a_{k}^{2}.}}$ 3) compare the result with P, if smaller, increase λ and go back to step 2) until finally $\mathcal{P} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}a_{k}^{2}}}$ with predefined accuracy By properly defining the range of λ and with some well-known search algorithms, the number of searches can be reduced significantly. A modified CF algorithm might possibly be needed if the channel changes very fast. Since decision errors were ignored, the actually achieved SIR will be lower than the expected, which equals $\frac{(\lambda)^{\prime}}{\sigma^{2}} - 1.$ Therefore, the following expression is a BER lower bound (LB) for MMSE-SIC receiver with the equal BER PC ${BER}_{LB} = {{E_{H}\left\lbrack {Q\left( \sqrt{\frac{(\lambda)^{\dagger}}{2} - 1} \right)} \right\rbrack}.}$

Simulation Results and Discussions

An indoor Rayleigh fading channel model is employed for simulations, with 100 MHz total bandwidth and τ_(d)=25 ns. The sub-carrier number N is chosen to be 16. Orthogonal Walsh Hadamard codes are employed for spreading. For each user, the instantaneous channel is randomly chosen from an ensemble consisted of 1000 i.i.d. Rayleigh fading channels (assumed unchanged for 100 symbols).

To emphasize the performance improvement with the proposed scheme, we compare in FIG. 2 the average BER performance, with different receiver structures, with and without PC, over 16 users versus the average E_(b)/N₀ per user. From this figure, we can see that the performance of MF (no SIC) ((a)) and MF-SIC with equal received power ((b)) are heavily limited by MAI, while MMSE (no SIC) ((d)) handles MAI much better than MF ((a) or (b)). Even with equal received power, employing SIC to MMSE ((e)) results in a significant performance improvement. Nevertheless, at a BER of 10⁻⁴, it is about 10 dB worse than the SUB ((h), equation (6)). Integrating MMSE-SIC with the proposed equal BER PC ((f)), additional 8 dB improvement can be obtained at a BER of 10⁻⁴, which is only less than 2 dB worse than the SUB, and it significantly outperforms MF-SIC with the equal BER PC ((c)). Moreover, it is interesting to note that the performance difference between the simulation result with MMSE-SIC with the equal BER PC ((f)) and the LB ((g), equation (13)) is very small, especially at high E_(b)/N₀, which justifies the assumption of ignoring decision errors.

FIG. 3 shows the received power allocation (averaged over 1000 channels, σ²=1) on 16 successive detected users. Not surprisingly, under different E_(b)/N₀, earlier detected users (larger index) are always allocated more power than the later detected ones (smaller index).

Under equal BER criterion, the PC algorithm disclosed for the MMSE-SIC receiver and its performance is thus analyzed and compared with other receiver strategies with and without PC in frequency-selective Rayleigh fading channels. From the results, we conclude that MMSE-SIC integrated with the equal BER PC is a powerful solution for suppressing MAI in uplink MC-CDMA systems.

Appendix A

Proof of Property that: a_(k) ²ε[0, +∞) (k=1, 2, . . . , K) are monotonically increasing

with λε[σ²,+∞)

Proof: Clearly, when λ=σ², a_(k) ²=0 (k=1, 2, . . . , K). When ignoring decision errors, the k^(th) detected symbol is only interfered by those haven't been detected ((k+1)^(th), (k+2)^(th), . . . , K^(th)) and its SIR can be expressed alternatively as SIR_(K-k+1)=a_(K-k+)1²{tilde over (C)}_(K-k+)1^(H)S_(K-k+)1⁻¹{tilde over (C)}_(K-k+)1, where $S_{K - k + 1} = {{\sum\limits_{j < k}{{\overset{\sim}{C}}_{K - j + 1}a_{K - j + 1}^{2}{\overset{\sim}{C}}_{K - j + 1}^{H}}} + {\sigma_{n}^{2}I}}$ and X_(k) denotes the k^(th) column of matrix X. For the last (K^(th)) detected symbol, since all interference has been perfectly cancelled, ${SIR}_{1} = {{\frac{\lambda}{\sigma^{2}} - 1} = {\frac{a_{1}^{2}{\overset{\sim}{C}}_{1}^{H}{\overset{\sim}{C}}_{1}}{\sigma^{2}}.}}$ Clearly, a₁ ² is monotonically increasing with λ. In another word, with λ₁>λ₂, a_(1|λ) ₁ ²>a_(1|λ) ₂ ². For the second last ((K−1)^(th)) detected symbol, ${{SIR}_{2} = {{\frac{\lambda}{\sigma^{2}} - 1} = {a_{2}^{2}{\overset{\sim}{C}}_{2}^{H}S_{2}^{- 1}{\overset{\sim}{C}}_{2}}}},$ where S₂={tilde over (C)}₁a₁ ²{tilde over (C)}₁ ^(H)+σ²I. When λ₁>λ₂, a_(1|λ) ₁ ²>a_(1|λ) ₂ ², hence, S_(2|λ) ₁ −S_(2|λ) ₂ is positive definite, which means S_(2|λ) ₁

S_(2|λ) ₂ . Obviously, (S_(2|λ) ₁ )⁻¹

(S_(2|λ) ₂ )⁻¹, thus, {tilde over (C)}₂ ^(H)((S_(2|λ) ₁ )⁻¹−(S_(2|λ) ₂ )⁻¹){tilde over (C)}₂<0. If a_(2|λ) ₁ ²≦a_(2|λ) ₂ ², ${a_{2|\lambda_{1}}^{2} \leq a_{2|\lambda_{2}}^{2}},{{\frac{\lambda_{1}}{\sigma^{2}} - 1} \leq {\frac{\lambda_{2}}{\sigma^{2}} - 1}},$ which conflicts with λ₁>λ₂. Therefore, to achieve a higher SIR (larger λ), a₂ ² must be increased to compensate for higher interference, which means, a₂ ² is also monotonically increasing with λ. Similar analysis can be made successively for the other symbols.

While the present invention has been described in terms of specific embodiments thereof, it will be understood in view of the present disclosure, that numerous variations upon the invention are now enabled to those skilled in the art, which variations yet reside within the scope of the present teaching. Accordingly, the invention is to be broadly construed, and limited only by the scope and spirit of the claims now appended hereto. 

1. A method for efficiently suppressing multiple access interferences (MAI) in an uplink MC-CDMA system, comprising integrating the nonlinear MMSE-SIC receiver in said system with an equal BER power control (BER PC).
 2. A method in accordance with claim 1 wherein the said system considers slow fading channels and a given decision order, for which, the MMSE-SIC receiver maximizes all users' signal-to-interference ratios (SIRs) simultaneously.
 3. A method in accordance with claim 2 wherein based on the channel state information (CSI) obtained at the receiver, the transmit power allocation of different users is calculated by use of a successive algorithm which searches the transmit power of different users under equal BER criterion with a total transmit power constraint.
 4. A method in accordance with claim 3 wherein the calculated power allocation is fed back to the transmitter so that each user will transmit with the assigned power.
 5. A method in accordance with claim 3 wherein the power allocation a_(k) ², is expressed in general successive form as $\begin{matrix} \left\{ \begin{matrix} {a_{1}^{2} = \frac{\lambda - \sigma^{2}}{r_{1,1}}} \\ {{a_{k}^{2} = {\frac{\lambda - \sigma^{2}}{r_{k,k}{\sum\limits_{j = 1}^{k - 1}{{\gamma_{j,k}}^{2}a_{j}^{- 2}\lambda}}}{\left( {{k = 2},\ldots\quad,K} \right).}}},} \end{matrix}\quad \right. & {{Equation}\quad(A)} \end{matrix}$ where K is the number of active users; where σ² denotes the noise variance; r_(i,j) denotes the (i,j)^(th) element of R, where R is the channel-modified cross correlation matrix; γ_(i,j) denotes the (i,j)^(th) element of Γ, which is obtained from the Cholesky factorization (CF) of the positive definite matrix R_(m)=R+σ²A⁻², where A=diag(a₁, a₂, . . . ,a_(K)) is a diagonal matrix containing the received amplitudes of all users; the said CF enabling R_(m) to be uniquely decomposed to be R_(m)=Γ^(H)D²Γ, where Γ is upper triangular and monic (having all ones along the diagonal) and D² is diagonal with all positive elements; and wherein the algorithm is implemented by the steps of 1) letting λ=σ²; 2) applying Equation (A) to calculate ${\frac{1}{K}{\sum\limits_{k = 1}^{K}{a_{k}^{2}.}}};$ 3) comparing the result with the power constraint P, and if smaller, increasing λ and going back to step 2) until finally $\overset{\_}{P} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}\quad a_{k}^{2}}}$ with predefined accuracy. 